THURSDAY, November 19th, 2009 at 4.10pm, MSB 1147 (Colloquium Room)
Refreshments: 3.30pm, MSB 4110 (Statistics Lounge)
Speaker: Naoki Saito (Dept. of Mathematics, UC Davis)
Title: Laplacian eigenfunctions that do not feel the boundary: Theory, Computation, and Applications
Abstract: I will discuss Laplacian eigenfunctions defined on a Euclidean domain of general shape, which "do not feel the boundary," how to compute them in practice, and their applications including image approximation and statistical image analysis. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz equation (which can be quite complicated and costly), we find the integral operator commuting with the Laplacian and diagonalize that operator.
Although our eigenfunctions satisfy neither the Dirichlet nor the Neumann boundary condition, computing the eigenfunctions via the integral operator is simple and has a potential to utilize modern fast algorithms (such as the Fast Multipole Method) to accelerate the computation. We also use our Laplacian eigenfunctions as a statistical image analysis tool, compare its performance with the standard Principal Component Analysis, and demonstrate that our tool can separate the statistics of data from the geometry of the domain where the data are supported, which is extremely difficult to achieve using PCA.
